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Mirrors > Home > MPE Home > Th. List > fusgrvtxfi | Structured version Visualization version GIF version |
Description: A finite simple graph has a finite set of vertices. (Contributed by AV, 16-Dec-2020.) |
Ref | Expression |
---|---|
isfusgr.v | ⊢ 𝑉 = (Vtx‘𝐺) |
Ref | Expression |
---|---|
fusgrvtxfi | ⊢ (𝐺 ∈ FinUSGraph → 𝑉 ∈ Fin) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | isfusgr.v | . . 3 ⊢ 𝑉 = (Vtx‘𝐺) | |
2 | 1 | isfusgr 29203 | . 2 ⊢ (𝐺 ∈ FinUSGraph ↔ (𝐺 ∈ USGraph ∧ 𝑉 ∈ Fin)) |
3 | 2 | simprbi 495 | 1 ⊢ (𝐺 ∈ FinUSGraph → 𝑉 ∈ Fin) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1533 ∈ wcel 2098 ‘cfv 6549 Fincfn 8964 Vtxcvtx 28881 USGraphcusgr 29034 FinUSGraphcfusgr 29201 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-ext 2696 |
This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-sb 2060 df-clab 2703 df-cleq 2717 df-clel 2802 df-rab 3419 df-v 3463 df-dif 3947 df-un 3949 df-ss 3961 df-nul 4323 df-if 4531 df-sn 4631 df-pr 4633 df-op 4637 df-uni 4910 df-br 5150 df-iota 6501 df-fv 6557 df-fusgr 29202 |
This theorem is referenced by: fusgrfupgrfs 29216 nbfusgrlevtxm1 29262 nbfusgrlevtxm2 29263 nbusgrvtxm1 29264 uvtxnm1nbgr 29289 cusgrm1rusgr 29468 wlksnfi 29790 fusgrhashclwwlkn 29961 clwwlkndivn 29962 fusgreghash2wsp 30220 numclwwlk3lem2 30266 numclwwlk4 30268 |
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